Let G be a group which is hyperbolic relative to a collection of subgroups H1, and it is also hyperbolic relative to a collection of subgroups H2. Suppose that H2 ⊂ H1. We characterize, for subgroups of G, when quasiconvexity relative to H1 implies quasiconvexity relative to H2. We also show that quasiconvexity relative toH2 implies quasiconvexity relative toH1. As an application, we give some ...

In the quadratic family (the set of polynomials of degree 2), Pe-tersen and Zakeri [PZ] proved the existence of Siegel disks whose boundaries are Jordan curves, but not quasicircles. In their examples, the critical point is contained in the curve. In the first part, as an illustration of the flexibility of the tools developped in [BC], we prove the existence of examples that do not contain the ...

Let θ : A → B be a zero-product preserving bounded linear map between C*-algebras. Here neither A nor B is necessarily unital. In this note, we investigate when θ gives rise to a Jordan homomorphism. In particular, we show that A and B are isomorphic as Jordan algebras if θ is bijective and sends zero products of self-adjoint elements to zero products. They are isomorphic as C*-algebras if θ is...

Lemma 3. The stability parameter of a performance measure Ψ(·) can be written as δ() ≤ L Ψ · iff its sufficient dual region is bounded in a ball of radius Θ (L Ψ). Proof. Let us denote primal variables using the notation x = (u, v) and dual variables using the notation θ = (α, β). The proof follows from the fact that any value of θ for which Ψ * (θ) = −∞ can be safely excluded from the sufficie...

1. Proofs of Theorems 1 and 2. Let η = (η T 1 , · · · , η T p) T be a (pq n)-vector and Θ = (Θ 1 , · · · , Θ p) be an n × (pq n) matrix. With matrix notation, the linear FAR criterion minimizes the following objective function

Let k be a field, T an algebraic k-torus, X smooth k-compactification of and PicX‾ the Picard group X‾=X×kk‾. Hoshi, Kanai Yamasaki [HKY22] determined H1(k,PicX‾) for norm one tori T=RK/k(1)(Gm) gave necessary sufficient condition Hasse principle extensions K/k number fields with [K:k]=n≤15 n≠12. In this paper, we determine 64 cases H1(k,PicX‾)≠0 give [K:k]=12.

Let (G,μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G,μ)-stationary space, with respect to the...

Let A =< A,≤A> and B =< B,≤B> be lattices such that A ∩ B is a filter in A and an ideal in B, and the orderings ≤A and ≤B coincide on A ∩ B. Then ≤A ∪ ≤B ∪ (≤A ◦ ≤B), is a lattice ordering on A ∪ B and the resulting lattice, called a sum of A and B, is denoted by A ⊕ B. The sum operation was introduced by Wroński [5], and its special case with A∩B = {1A} = {0B} by Troelstra [4]. In particular, ...

For (b), let Θ be any subset of Γ, x in its complement, U as in (a). The neighbourhood xU of x contains at most one element of Γ. There exists a neighbourhood of x contained in xU and not containing any element of Θ. For (c), let V be the closure of U ·U, which is compact. The intersection of Γ with V is compact, and covered by disjoint neighbourhoods of each of its points. This intersection mu...

Let C be a smooth, connected, projective non-hyperelliptic curve of genus g ≥ 2 over the complex numbers and let Pic(C) be the connected component of its Picard variety parametrizing degree d line bundles, for d ∈ Z. The variety Pic(C) carries a naturally defined divisor, the Riemann theta divisor Θ, whose support consists of line bundles that have nonzero global sections. The Riemann singulari...